Periodic Solutions Of Delay Differential Equations Research Articles

Parametric Sensitivity Analysis of Oscillatory Delay Systems with an Application to Gene Regulation.

A parametric sensitivity analysis for periodic solutions of delay-differential equations is developed. Because phase shifts cause the sensitivity coefficients of a periodic orbit to diverge, we focus on sensitivities of the extrema, from which amplitude sensitivities are computed, and of the period. Delay-differential equations are often used to model gene expression networks. In these models, the parametric sensitivities of a particular genotype define the local geometry of the evolutionary landscape. Thus, sensitivities can be used to investigate directions of gradual evolutionary change. An oscillatory protein synthesis model whose properties are modulated by RNA interference is used as an example. This model consists of a set of coupled delay-differential equations involving three delays. Sensitivity analyses are carried out at several operating points. Comments on the evolutionary implications of the results are offered.

A new Ф-generalized quasi metric space with some fixed point results and applications

In this paper, the concept of a new ?-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from ?-generalized quasi metric space. Some general fixed point theorems in a ?-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.

Periodic solutions of delay-differential equations with a restorative condition

Periodic solutions of delay-differential equations with a restorative condition

Oscillatory periodic solutions of differential-delay equations with multiple lags

BASED on the Nussbaums results, we studied in ref. [2] the conditions for the existence of oscillatory periodic solutions to the differential-delay equations (DDEs)...

Periodic solutions of differential-delay equations with more than one delay

On demontre l'existence de solutions periodiques non triviales pour certaines equations differentielles a retard avec plus d'un retard

Almost periodic solutions for delay-differential equations with infinite delays

One of the most widely methods for establishing the existence of almost periodic (a.p. for short) solutions of systems or ordinary differential equations is based on a general result due to Amerio [ 1 J, a generalization of an earlier result for linear a.p. systems due to Favard. In essence, this basic method consists of showing that if a solution is in a compact set in the statespace for all t and has a certain separation property with respect to other such solutions, then if this same property holds for all systems in the socalled hull of the given system, this solution will be a.p. This result in fact relates to a necessary as well as sufficient stability condition for a fairly general dynamical system to have an a.p. solution; cf. [2, Chap. 51. In fact Miller [3] used the idea of embedding the a.p. system in a more general dynamical system and was thus able to obtain a local stability condition for the existence of a.p. solutions without using the separation conditions due to Favard and Amerio. Most stability conditions for the existence of a.p. solutions known prior to Miller’s result were in terms of global stability conditions. For a fairly complete and comprehensive discussion of a.p. solutions for a.p. systems, the books by Fink [4] and Yoshizawa [5] are recommended. For a more general discussion of the idea of embedding a.p. systems in dynamical systems, cf. Sell [6]. For ordinary differential equations with fixed finite time delays, much of the method based on Amerio’s separation result can and has been adapted to obtain similar stability conditions for the existence of a.p. solutions; cf. [7], for example. Miller in [3] shows that local stability conditions can also be used for such delay-differential equations. Results for delay equations with infinite time delays based on Amerio’s separation condition have also been obtained; cf., for example, Hino [B]. However, for such systems which do not involve substantially fading memory, not much seems to have been done; one of the main difficulties in such infinite delay systems arises from the fact that bounded solutions may no longer remain in compact subsets of the state

A Perturbative Approach to Periodic Solutions of Delay-differential Equations

A Perturbative Approach to Periodic Solutions of Delay-differential Equations